ANALYSIS OF STRAIN LOCALIZATION BY ENERGY EQUIVALENCE: Ⅰ. ONE-DIMENSIONAL ANALYTICAL SOLUTION

Based on the first law of thermodynamics and the nonlocal plasticity theories, a new approach is proposed to solve the strain localization problems induced by strain softening. For each material point, two state spaces, local and nonlocal state spaces, are defined such that the local internal variable can be mapped, from the local state space by integral transformation with the nonlocal weighting function, into the nonlocal internal variable in the nonlocal state space. During strain softening, the plastic deformation follows the normal flow rule in the local state space and the softening law is introduced in the nonlocal state space. It is assumed that the strain softening is a global material behavior and the plastic energy dissipation within the entire material body is always positive. However, the balance of momentum is still satisfied locally. By equating the rates of the plastic energy dissipation in the two state spaces during strain softening, the localization zone and the plastic strain distribution become well-defined. Analytical solution for the one-dimensional strain localization is developed, and it is well shown that the plastic strain distribution and load-displacement curves are well-defined by the material properties, such as the softening modulus and internal length scale, as well as the geometry of the material body. For the Gaussian-type weighting functions the width of the localization zone is approximately six times the internal length scale. Numerical example demonstrates that the size of the localization zone decreases as the internal length scale is reduced, and the distribution of the plastic strain in the localization zone becomes more concentrated when the internal length scale becomes smaller. As the internal length scale approaches to zero, the solution reduces to the one predicted by the conventional local plasticity theory.